Kardar-Parisi-Zhang scaling in time-crystalline matter

Armin Asadollahi; Carl Philipp Zelle; Romain Daviet; Sebastian Diehl

REVIEW 2 major objections 5 minor 3 cited by

Kardar-Parisi-Zhang scaling in time-crystalline matter

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The Goldstone mode of broken time-translation symmetry—the 'chronon' of a limit-cycle phase—is governed by a compact Kardar-Parisi-Zhang equation, making KPZ scaling the universal fate of time crystals and related nonequilibrium phases.

desk verdict A clean symmetry argument that time-crystal Goldstone modes generically obey compact KPZ, with a concrete Van der Pol derivation and numerical support; the abstract overstates the robustness, but it deserves refereeing. read the letter →

arxiv 2412.09677 v2 pith:2IOHRIAJ submitted 2024-12-12 cond-mat.stat-mech

Romain Daviet , Carl Philipp Zelle , Armin Asadollahi , Sebastian Diehl This is my paper · ORCID

classification cond-mat.stat-mech

keywords timecrystalslimit-cyclephasesGoldstonemodeKardar-Parisi-ZhangequationcompactKPZuniversalityVanderPoloscillatorsspontaneoustime-translationsymmetrybreakingdriven-dissipativesystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that whenever a many-body system spontaneously breaks continuous time-translation symmetry—entering a limit-cycle or continuous-time-crystal phase—the soft fluctuation mode along the cycle, which the authors call the chronon, obeys the Kardar-Parisi-Zhang (KPZ) equation. Because the chronon is an angular variable and the phase keeps growing around the cycle, the KPZ nonlinearity is the only symmetry-allowed interaction, so it is generated generically. This puts an enormous class of nonequilibrium systems—synchronized oscillators, nonreciprocal active matter, active magnets, driven-dissipative condensates—into a single universality class with nontrivial scaling: in one dimension the exact KPZ exponents, in two dimensions the approximate ones, with topological vortex defects cutting off order at large scales. The authors verify the prediction in simulations of the Van der Pol oscillator in one and two dimensions, matching the KPZ exponents.

What carries the argument

The central object is the chronon, the compact $SO(2)$ Goldstone mode that describes local, slowly varying phase shifts along a limit cycle. The machine that carries the argument is the symmetry-constrained effective action: time-translation invariance forces invariance under $\theta\to\theta+c$, the continuous growth of the phase forbids $\theta\to-\theta$, and the only leading interaction allowed is the KPZ term $\tilde{\theta}(\nabla\theta)^2$. Combined with adiabatic elimination of the gapped transverse modes and averaging over one period, this leaves the single-component KPZ action of Eq. (4); no internal continuous symmetry is required.

What would settle it

In a one-dimensional chain of coupled noisy Van der Pol oscillators, measure the envelope of the order-parameter autocorrelation $C(t)$: KPZ predicts $-\ln C(t,0) \sim A t^{2/3}$; observing $t^{1/2}$ diffusive scaling, or an exponential cutoff at scales far smaller than the predicted vortex scale, would falsify the central claim.

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Extended reading notes

Core claim

Starting from a generic functional-integral action for a driven open system whose order parameter traces a limit cycle $\varphi(t)$, the authors promote the global time shift to a slow, space-dependent field $\theta(t,x)$ and parameterize fluctuations as $\varphi(t+\bar{T}\theta(t,x)) + N(t,x)$. After integrating out the gapped transverse fluctuations $N$ and averaging over the limit-cycle period $\bar{T}$, the effective action for the chronon is $S=\int_{t,x} \tilde{\theta}\big(\partial_t\theta - Z\nabla^2\theta + \tfrac{g}{2}(\nabla\theta)^2\big) - D\tilde{\theta}^2$, which is precisely the action of the KPZ equation. The structural point is that a constant shift $\theta\to\theta+c$ is a symmetry whereas reflection $\theta\to-\theta$ is not, so the KPZ coupling $g$ is symmetry-allowed and inevitably generated, and it is forbidden for ordinary spatial or internal symmetry breaking. Because the chronon is an angle, space-time vortices are allowed, so correlations follow KPZ stretched-exponential decay on intermediate scales and exponential decay beyond the vortex scale $L_v$. The explicit Van der Pol reduction yields this KPZ action, and simulations in one and two dimensions reproduce the predicted exponents: $\beta=0.31$ versus the exact $1/3$ in $d=1$, and $\beta=0.24$, $2\chi\approx0.76$ versus $\chi\approx0.39$ in $d=2$.

Load-bearing premise

The argument holds if the fast, gapped fluctuations away from the limit cycle can be integrated out and the explicitly time-periodic coefficients averaged over one cycle without introducing additional relevant operators, leaving exactly the single-component KPZ action.

Editorial extensions

If this is right

In every continuous time crystal and generic limit-cycle phase in $d=1$ and $d=2$, the order-parameter correlations decay as KPZ stretched exponentials on scales below the vortex separation, with $\beta=1/3$, $\chi=1/2$ in $d=1$ and $\beta\approx0.24$, $\chi\approx0.39$ in $d=2$.
Platforms with no broken internal continuous symmetry—arrays of Van der Pol oscillators, nonreciprocal active matter, active magnets, synchronized oscillators—are predicted to display KPZ scaling purely from time-translation breaking.
At higher noise levels, topological defects (space-time vortices) proliferate and produce exponential decay of correlations beyond a scale $L_v \sim e^{\Delta/\sigma}$, so the KPZ regime is a low-noise, intermediate-scale phenomenon.
Traveling-wave states and coherently driven condensates, where internal symmetries are reduced to discrete subgroups, are predicted to host chronon modes with KPZ or anisotropic-KPZ scaling.
The framework unifies previously observed KPZ behavior in exciton-polariton condensates and oscillator lattices as manifestations of time-translation symmetry breaking rather than as special features of those models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Going beyond the paper: if the chronon description is exact, exact one-dimensional KPZ results—universal distribution functions and correlation functions—should appear in the fluctuations of synchronized oscillator arrays and could be measured in room-temperature classical experiments.
Going beyond the paper: in $d=2$ the theory predicts a crossover from KPZ scaling to vortex-dominated decay; a quantitative measurement of $L_v$ as a function of noise strength would test the compactness assumption independently of the KPZ exponents.
Going beyond the paper: the framework suggests that in $d=3$, where KPZ has a roughening transition, time-crystalline phases could be used to tune through this transition by varying noise or coupling, giving access to a nonequilibrium phase transition in a synthetic setting.
Going beyond the paper: when the Goldstone mode is coupled to conserved densities, as in nonreciprocal phase-separation models, conserved KPZ variants may replace standard KPZ; distinguishing these universality classes in simulations would delineate the boundary of the claim.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here’s the short version: the paper’s core claim—that the phase Goldstone mode of any continuous time crystal is governed by compact KPZ—is likely right in the generic case, and the authors back it with an explicit Van der Pol reduction and numerical checks. It deserves refereeing, but the abstract oversells the robustness, and the referee should push on the two uncontrolled steps in the effective field theory derivation.

What’s genuinely new: the symmetry argument tying the 'chronon' (the Goldstone mode of broken time translations) to compact KPZ, unifying known KPZ results in polariton condensates, driven condensates, oscillator lattices, and now active matter and nonreciprocal magnets. Prior works established KPZ in specific models; this paper gives a common mechanism. The explicit Van der Pol derivation in the SM is a real contribution: they project onto tangent and normal modes, average the fast period, integrate out the gapped normal mode, and arrive at KPZ with coefficients computed from microscopic parameters. Not circular. The d=1 and d=2 simulations match KPZ exponents, and code/data are on Zenodo. Credit where due.

Soft spots, in order of importance. First, the general claim is broader than the derivation proves. The separation of scales is stated—slow phase, fast period, gapped transverse mode—and the Van der Pol calculation is explicit, but the corrections from integrating out N are not computed, and the averaging of the noise vertex is asserted. The paper does flag in the SM that conserved or noisy variants of the Goldstone mode would change the class, and the conclusion notes that additional soft modes from conserved quantities can couple in. That is the right caveat, but it sits in the SM and conclusion, not in the abstract, which reads as a blanket prediction. A referee should ask for a more precise statement of when the standard KPZ class applies and when it doesn’t. Second, the exponent fits have no error bars; the values are consistent with KPZ (β=0.31 vs 1/3 in d=1; β=0.24 and 2χ≈0.76 vs ≈0.78 in d=2), but a few error bars would make it solid. Third, the vortex-dominated regime is shown but not quantitatively analyzed; the claim that defects destroy long-range order in d=1,2 is standard, but the crossover scale is not connected to the KPZ regime.

One more claim deserves scrutiny: the statement that the KPZ nonlinearity is forbidden for every other symmetry-breaking pattern, in and out of equilibrium. It’s plausible—the limit cycle’s growth direction breaks θ → −θ—but it is stated without proof. A counterexample from an internal U(1) breaking in a stationary driven condensate would be worth checking.

Who gets value: the time-crystal and active-matter communities, and anyone trying to classify nonequilibrium Goldstone dynamics. It’s a useful organizing paper. I’d send it to a serious referee, with instructions to focus on the derivation’s assumptions and to ask the authors to temper the abstract.

Referee Report

2 major / 5 minor

Summary. The paper argues that the Goldstone mode of spontaneously broken continuous time-translation symmetry in limit-cycle phases—dubbed the 'chronon'—is governed by a compact Kardar-Parisi-Zhang (KPZ) equation. Starting from a Keldysh/MSRJD path integral, the authors propose a symmetry-based effective action for the phase field and claim that, after period averaging and adiabatic elimination of gapped transverse fluctuations, the single-component KPZ action of Eq. (4) follows. They then predict KPZ scaling for the phase correlator, including the known d=1 and d=2 exponents, and complement the derivation with numerical simulations of an extended Van der Pol model in one and two dimensions. The paper also lists several physical platforms—nonreciprocal active matter, driven-dissipative condensates, active magnets, and oscillator arrays—where the chronon mechanism should apply. The central claim is presented as a robust symmetry-based universality statement, with the explicit Van der Pol calculation relegated to the Supplemental Material.

Significance. If the reduction to the compact KPZ equation is correct, the paper identifies a genuinely general mechanism: because time translation is an exact external symmetry, the resulting Goldstone mode is protected against many microscopic perturbations that would destroy internal symmetry-based mechanisms. The paper gives credit where due: the Van der Pol coefficients Z, g, and D are computed from the microscopic parameters rather than fitted, the predicted exponents are taken from independent KPZ results, and the numerical code and data are made available on Zenodo. The d=1 and d=2 simulations, while not error-bounded, are consistent with the claimed exponents. The manuscript also correctly distinguishes its own regime of validity by noting that conserved or noisy variants of the Goldstone mode would change the universality class. The significance would be high if the uncontrolled steps in the reduction were replaced by a controlled argument; as it stands, the general claim is plausible but not fully established.

major comments (2)

[SM III.A, Eq. (S7)] The replacement of the explicitly periodic coefficients Z[t+θT̄], D[t+θT̄], and g[t+θT̄] in Eq. (S7) by their period averages is the decisive step that turns the time-quasiperiodic action into the time-translation-invariant KPZ action of Eq. (4). The manuscript states that this can be done 'safely' because τ≫T̄, but no controlled expansion is provided: there is no explicit small parameter, no estimate of the corrections of order T̄/τ, and no discussion of how the θ-dependence inside t+θT̄ affects the averaging. This is load-bearing because if the averaging fails, additional θ-dependent or time-dependent operators can become relevant and the universality class changes. The caveat in SM III.A about conserved noise is a step in the right direction, but the main text's general claim requires a controlled derivation rather than a formal statement.
[SM III.B, Eqs. (S12)-(S14)] The adiabatic elimination of the gapped transverse mode N is asserted but not actually performed. The text says that solving Eq. (S12) for N(θ) is 'equivalent to performing the Gaussian integral over N', but the result of that integral is never shown. In particular, the noise vertex D_N and the couplings Y_N and g_N in Eqs. (S12) and (S13) can generate a non-Markovian or conserved-noise contribution to the effective θ dynamics unless specific cancellations occur. Without the explicit integrated action, the claim that Eq. (8) has white noise and the standard KPZ coefficient g is not established. This issue is load-bearing because a conserved-noise component would place the system in a different universality class (cf. Refs. [121,122] and the caveat in SM III.A).

minor comments (5)

[Abstract] There are typos in the abstract: 'predicts an rationalizes' should be 'predicts and rationalizes', and 'it occur' should be 'it can occur'.
[SM III.B, Eq. (S10)] The normalization denominator in Eq. (S10) appears to be incorrect. Since φ=(f, ḟ), the tangent vector should be normalized by √(ḟ²+f¨²), not by f²+ḟ² as written; as printed, ê_θ and ê_N are not unit vectors.
[Figs. 2 and 3] The fitted exponents (β=0.31 in d=1; 2χ≈0.76 and β≈0.24 in d=2) are reported without error bars. Reporting the fit range, the number of independent samples, and an uncertainty estimate would strengthen the 'excellent accuracy' claim.
[Main text, Eq. (5)] In Eq. (5), the 'last line' retains only the leading harmonic contribution to the envelope; it would be clearer to state explicitly that the leading scaling envelope is dominated by the lowest harmonic.
[Data availability] The data and code statement refers to Zenodo but does not provide a DOI or URL; a working link should be included.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the compact-KPZ mapping is derived from the microscopic Van der Pol action by period averaging and adiabatic elimination, with exponents taken from independent KPZ literature; self-citations are contextual, not load-bearing.

full rationale

The derivation chain is not circular. The paper starts from a microscopic Van der Pol field theory (Eq. 7), parametrizes fluctuations as phi(t + Tbar theta) + N (Eq. 3), projects the Langevin dynamics onto the tangent and normal directions (SM Eqs. S12-S13), and then obtains the KPZ action (Eqs. 4 and 8) by period averaging and adiabatic elimination, with Z and g expressed through integrals of the limit-cycle solution (SM Eq. S14). No parameter appearing in the final KPZ equation is fitted to the quantity being predicted; the scaling exponents beta = 1/3, chi = 1/2 in d = 1 and beta ~ 0.24, chi ~ 0.39 in d = 2 are taken from independent KPZ results (Refs. 88-92). The numerical simulations solve the full Van der Pol equation (7) and compare the extracted exponents to these independent values, so the comparison is external. Self-citations such as Refs. [33,50] for the Goldstone theorem, [57,58] for compact KPZ in condensates, and [60,61,93] for vortex physics are present, but the Goldstone theorem is an established result also cited to independent work (Refs. 48,49,51), and the compact-KPZ behavior has external experimental support (Refs. 72,104); none of these citations is the sole justification for the central mapping. The uncontrolled step is the period averaging and elimination of the gapped mode N (SM Section III.B), which the paper itself flags as mechanism-dependent by noting that conserved or noisy variants would yield different KPZ variants (SM after Eq. S7). That is a correctness or robustness concern, not circularity, because the final action is not logically identical to its inputs by construction. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central mapping introduces no fitted constants: microscopic coefficients (Z1, Z2, u, gamma, omega0, D) determine the KPZ coefficients Z, g, D via averages over the limit cycle. The main burden is the domain assumptions listed above, especially period averaging and adiabatic elimination. Numerical checks use the same model and known KPZ exponents, not fitted predictions.

assumptions (5)

standard math Spontaneously broken continuous time-translation symmetry implies a gapless Goldstone mode ('chronon') in driven open systems.
Invoked in the Introduction and SM Eqs. (S4, S5); relies on the Goldstone theorem for time translations adapted to the Keldysh framework, citing Refs. 33, 48-51.
domain assumption Time-averaging over a limit-cycle period is valid for slowly varying Goldstone fluctuations (tau >> T).
Used to pass from Eq. (S7) to the time-independent KPZ action Eq. (4); breaks down for fast fluctuations or near bifurcations.
domain assumption Gapped transverse fluctuations N can be adiabatically eliminated without generating further relevant operators.
SM around Eqs. (S12)-(S14); if elimination generated conserved noise or extra soft modes, the single-component KPZ universality class could change.
domain assumption Topological defects of the compact phase are always activated in d=1 and d=2, with L_v ~ exp(Delta/sigma).
Used to predict exponential decay beyond a scale L_v; taken from Refs. 60, 61, 93 and assumed generic for limit-cycle Goldstone modes.
standard math Known KPZ exponents in d=1 and d=2 are accurate and apply to the compact KPZ equation at intermediate scales.
Used in Eq. (6) and for comparison with simulations; references Refs. 63 and 85-92.

invented entities (1)

Chronon: compact SO(2) Goldstone mode of time-translation symmetry breaking independent evidence
purpose: Collective phase fluctuation along the limit cycle, proposed to obey compact KPZ dynamics.
Although the mode is the standard phase fluctuation of a limit cycle, this paper names it and assigns a falsifiable universal signature: KPZ scaling with specific exponents and a vortex crossover that can be measured in correlation functions of several proposed platforms.

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Cite this review

Pith. "Pith review of Kardar-Parisi-Zhang scaling in time-crystalline matter." pith.science (2026). https://pith.science/paper/2IOHRIAJ

@misc{pith2026241209677,
  author       = {Pith},
  title        = {Pith review of: Kardar-Parisi-Zhang scaling in time-crystalline matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2IOHRIAJ}},
  note         = {Machine review of arXiv:2412.09677}
}

read the original abstract

We discuss the universal behavior linked to the Goldstone mode associated with the spontaneous breaking of time-translation symmetry in many-body systems, in which the order parameter traces out a limit cycle. We show that this universal behavior is closely tied to Kardar-Parisi-Zhang physics, which can strongly affect the scaling properties in all dimensions. Our work predicts the relevance of KPZ in numerous systems such as nonreciprocal phases in active matter, active magnets, driven-dissipative quantum systems, and synchronization of oscillators.

Figures

Figures reproduced from arXiv: 2412.09677 by the authors.

**Figure 2.** FIG. 2. Autocorrelation function [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Simulations of the Van der Pol equation in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Two-dimensional chiral Malthusian flocks map to (2+1)-dimensional KPZ growth, so any chirality destroys long-range order but leaves a weak-chirality cascade of quasi-ordered regimes.

Reference graph

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The right bottom inset shows a space-time vortex of ( ϕ, ∂tϕ) at higher noise, D = 2

The autocorrelation function displays rapid oscillations (see inset), but its envelope (red line) shows the expected KPZ scaling at sufficiently large scales as shown through the fit for t ∈ [102, 103] (black dotted line), which yields β = 0.31, which is in very good agreement with the theoretical value. The right bottom inset shows a space-time vortex of...
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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

The right bottom inset shows a space-time vortex of ( ϕ, ∂tϕ) at higher noise, D = 2

The autocorrelation function displays rapid oscillations (see inset), but its envelope (red line) shows the expected KPZ scaling at sufficiently large scales as shown through the fit for t ∈ [102, 103] (black dotted line), which yields β = 0.31, which is in very good agreement with the theoretical value. The right bottom inset shows a space-time vortex of...

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